The quadratic polynomial p(x) has roots 1 and , while quadratic polynomial q(x) has roots 1 and . Let and be the roots of r(x) = p(x) + q(x) . Then _ x [ p(x) - q(x) ] =

Step-by-step solution

Let the quadratic polynomials be p(x) = a(x-1)(x- ) and q(x) = b(x-1)(x- ) . Given r(x) = p(x) + q(x) , we can evaluate r(x) at x = 1 : r(1) = p(1) + q(1) = 0 + 0 = 0 Since the roots of r(x) are given as and , and r(1) = 0 , one of the roots must be 1 . Without loss of generality, let = 1 . Then p(x) = a(x-1)^2 . Since the roots of r(x) are 1 and , we can write r(x) = (a+b)(x-1)(x- ) . Substitute p(x) , q(x) , and r(x) into r(x) = p(x) + q(x) : (a+b)(x-1)(x- ) = a(x-1)^2 + b(x-1)(x- ) Dividing by (x-1) for x 1 : (a